scholarly journals Convexity in complex networks

2018 ◽  
Vol 6 (2) ◽  
pp. 176-203 ◽  
Author(s):  
TILEN MARC ◽  
LOVRO ŠUBELJ
Keyword(s):  
Complex Networks ◽  
Geodesic Distance ◽  
Connected Subgraph ◽  
Geodesic Path ◽  
Connected Subset ◽  
Social Collaboration ◽  
Graph Properties ◽  
Definition Of ◽  
Locally Convex ◽  
Convex Core

AbstractMetric graph properties lie in the heart of the analysis of complex networks, while in this paper we study their convexity through mathematical definition of a convex subgraph. A subgraph is convex if every geodesic path between the nodes of the subgraph lies entirely within the subgraph. According to our perception of convexity, convex network is such in which every connected subset of nodes induces a convex subgraph. We show that convexity is an inherent property of many networks that is not present in a random graph. Most convex are spatial infrastructure networks and social collaboration graphs due to their tree-like or clique-like structure, whereas the food web is the only network studied that is truly non-convex. Core–periphery networks are regionally convex as they can be divided into a non-convex core surrounded by a convex periphery. Random graphs, however, are only locally convex meaning that any connected subgraph of size smaller than the average geodesic distance between the nodes is almost certainly convex. We present different measures of network convexity and discuss its applications in the study of networks.

WEIGHTED AVERAGE GEODESIC DISTANCE OF PENTADENDRITE NETWORKS

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050075
Author(s):  
YUANYUAN LI ◽  
XIAOMIN REN ◽  
KAN JIANG
Keyword(s):  
Complex Networks ◽  
Geodesic Distance ◽  
Average Distance ◽  
Weighted Average ◽  
Similar Measure ◽  
Important Index ◽  
Self Similar

The average geodesic distance is an important index in the study of complex networks. In this paper, we investigate the weighted average distance of Pentadendrite fractal and Pentadendrite networks. To provide the formula, we use the integral of geodesic distance in terms of self-similar measure with respect to the weighted vector.

scholarly journals Convex skeletons of complex networks

2018 ◽  
Vol 15 (145) ◽  
pp. 20180422 ◽  
Author(s):  
Lovro Šubelj
Keyword(s):  
Protein Interactions ◽  
Spanning Tree ◽  
Autonomous Systems ◽  
Shortest Paths ◽  
Induced Subgraph ◽  
Social Collaboration ◽  
Computer Scientists ◽  
High Betweenness ◽  
Network Backbone ◽  
Definition Of

A convex network can be defined as a network such that every connected induced subgraph includes all the shortest paths between its nodes. A fully convex network would therefore be a collection of cliques stitched together in a tree. In this paper, we study the largest high-convexity part of empirical networks obtained by removing the least number of edges, which we call a convex skeleton. A convex skeleton is a generalization of a network spanning tree in which each edge can be replaced by a clique of arbitrary size. We present different approaches for extracting convex skeletons and apply them to social collaboration and protein interactions networks, autonomous systems graphs and food webs. We show that the extracted convex skeletons retain the degree distribution, clustering, connectivity, distances, node position and also community structure, while making the shortest paths between the nodes largely unique. Moreover, in the Slovenian computer scientists coauthorship network, a convex skeleton retains the strongest ties between the authors, differently from a spanning tree or high-betweenness backbone and high-salience skeleton. A convex skeleton thus represents a simple definition of a network backbone with applications in coauthorship and other social collaboration networks.

A network function-based definition of communities in complex networks

2012 ◽  
Vol 22 (3) ◽  
pp. 033129 ◽  
Author(s):  
Sanjeev Chauhan ◽  
Michelle Girvan ◽  
Edward Ott
Keyword(s):  
Complex Networks ◽  
Network Function ◽  
Definition Of

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

scholarly journals Path-integral formulation of spreading processes in complex networks

2021 ◽  
Author(s):  
Flavio Iannelli ◽  
Igor M. Sokolov
Keyword(s):  
Complex Networks ◽  
Path Integral ◽  
Disease Transmission ◽  
Arrival Time ◽  
Geodesic Distance ◽  
Integral Formulation ◽  
Numerical Estimation ◽  
Path Integral Formulation ◽  
Spreading Processes

AbstractWe introduce a path-integral formulation of network-based measures that generalize the concept of geodesic distance and that provides fundamental insights into the dynamics of disease transmission as well as an efficient numerical estimation of the infection arrival time.

scholarly journals A generalized inductive limit topology for linear spaces

1973 ◽  
Vol 14 (2) ◽  
pp. 105-110 ◽  
Author(s):  
S. O. Iyahen ◽  
J. O. Popoola
Keyword(s):  
Inductive Limit ◽  
Locally Convex Spaces ◽  
Closed Graph Theorem ◽  
Linear Maps ◽  
Inductive Limit Topology ◽  
Definition Of ◽  
Object Of Study ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Convex Spaces

In the usual definition of an inductive limit of locally convex spaces, one is given a linear space E, a family (Eα) of locally convex spaces and a set (iα) of linear maps from Eα into E. Garling in [2] studies an extension of this, looking at absolutely convex subsets Sα of Eα and restrictions jα of iα to such sets. If, in the definition of Garling [2, p. 3], each Sα is instead a balanced semiconvex set, then the finest linear (not necessarily locally convex) topology on E for which the maps ja are continuous, will be referred to as the generalized *-inductive limit topology of the semiconvex sets. This topology is our object of study in the present paper; we find applications in the closed graph theorem.

New definition of locally convex L-topological vector spaces

2009 ◽  
Vol 160 (9) ◽  
pp. 1245-1255 ◽  
Author(s):  
Hua-Peng Zhang ◽  
Jin-Xuan Fang
Keyword(s):  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Definition Of ◽  
Locally Convex

scholarly journals A Potential Information Capacity Index for Link Prediction of Complex Networks Based on the Cannikin Law

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 863 ◽  
Author(s):  
Xing Li ◽  
Shuxin Liu ◽  
Hongchang Chen ◽  
Kai Wang
Keyword(s):  
Complex Networks ◽  
Information Transmission ◽  
Link Prediction ◽  
Information Capacity ◽  
Prediction Methods ◽  
Transmission Process ◽  
Capacity Index ◽  
Definition Of ◽  
Definition Of Information ◽  
Transmission Capability

Recently, a number of similarity-based methods have been proposed for link prediction of complex networks. Among these indices, the resource-allocation-based prediction methods perform very well considering the amount of resources in the information transmission process between nodes. However, they ignore the information channels and their information capacity in information transmission process between two endpoints. Motivated by the Cannikin Law, the definition of information capacity is proposed to quantify the information transmission capability between any two nodes. Then, based on the information capacity, a potential information capacity (PIC) index is proposed for link prediction. Empirical study on 15 datasets has shown that the PIC index we proposed can achieve a good performance, compared with eight mainstream baselines.

scholarly journals Bounded generators in linear topological spaces

1971 ◽  
Vol 12 (2) ◽  
pp. 105-109
Author(s):  
S. O. Iyahen
Keyword(s):  
Banach Spaces ◽  
Convex Space ◽  
Locally Convex Space ◽  
Topological Spaces ◽  
Bounded Set ◽  
Definition Of ◽  
Bounded Sets ◽  
Locally Convex

Ito and Seidman in [5] define a BG space as a locally convex space in whichthere exists a bounded set with a dense span. In this note we extend the idea to a class of not necessarily locally convex linear topological spaces (l.t.s.). We note the link between the idea of a BG space and Weston’s characterization in [7] of separable Banach spaces. Finally we examine σ-BG spaces; here the bounded set in the definition of a BG space is replaced by the union of a sequence of bounded sets.

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